Chapter 37 - SANS FROM POLYMER SOLUTIONS

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1 Chapter 37 - SANS FROM OLYMER SOLUTIONS Soluility is a determining factor in the synthesis, mixing aility and end-use of polymers. A general model for descriing soluility (Flory, 1953) is discussed here with an emphasis on what information is otained from SANS measurements from polymer solutions. SANS data from specific polymer solutions are discussed in some detail. 1. OLYMER SOLUTIONS BASICS Most non-polar polymers dissolve in organic solvents and some polar polymers dissolve in water. Concentration ranges vary from dilute to semi-dilute to concentrated solutions. The orderline etween the dilute and the semi-dilute regimes is referred to as the overlap 3 concentration (c*) which is estimated as c * ~ M w /( 4π 3) R g (M w is the molecular weight and R g is the radius of gyration). The polymer concentration c is related to the volume fraction through the density d as c = d.. CASE OF A SIMLE OLYMER SOLUTION As an example of a polymer solution, the case of poly(ethylene oxide) EO in water is discussed (Hammouda et al, 00; Hammouda-Ho, 007). The EO monomeric unit - (CH CH O)- is the simplest one for a water-solule polymer. When dissolved in water, EO is characterized y hydrophilic interactions (hydrogen onding of water molecules to the oxygen atoms on the polymer) and hydrophoic interactions (the CH CH groups repel water). EO dissolves in water for a wide range of temperatures and concentrations. Its homologues, MO (-CH O-) and O (-CH(CH 3 )CH O-) do not dissolve in water an amient temperature. This may e due to the fact that the oxygen-oxygen inter-distance on the EO chain matches the oxygen-oxygen inter-distance in the structure of pure water. A typical SANS spectrum from a 4 % EO/d-water (weight average and numer average molecular weights of M w = 100,000 and M n = 96,000 g/mol respectively) is plotted for the T = 10 o C temperature. This sample is well in the semi-dilute region (c* is estimated to e g/cm 3 which corresponds to.4 % volume fraction). The low-q feature characterizes large size clusters (of no interest to us here) and the high-q feature characterizes the polymer chains. 1

2 4% EO/d-water, M w =100,000 g/mole, T=10 o C Scattering Intensity (cm -1 ) SANS Data Scattering Variale Q (Å -1 ) Figure 1: SANS data for a EO/d-water sample over a wide Q range showing a low-q feature and a high-q feature. Only the tail of the low-q feature is oserved. Focus here is on the high-q feature. 3. FIT TO A SIMLE MODEL In order to characterize our results, the following empirical functional form is fitted to the data: I(Q) A C = + + B. (1) n m Q 1+ ( Qξ) The first term descries orod scattering from clusters and the second term descries scattering from polymer chains. This second term characterizes the polymer/solvent interactions and therefore the thermodynamics and is of interest here. The two multiplicative factors A and C, the incoherent ackground B and the two exponents n and m are used as fitting parameters. The final parameter ξ is a correlation length for the entangled polymer chains. It gives an estimate of the entanglement length (average distance etween entanglements). Non-linear least squares fits to the empirical functional form yield ξ = 0 Å, and m = 1.9 for the 4 % EO/d-water sample. This empirical model

3 should e used with caution since it does not reproduce the Guinier limit properly (except for m = ). 4. THE CORRELATION LENGTH The correlation length ξ decreases with increasing polymer volume fraction ecause the entanglement length increases. ξ goes from close to 80 Å at low polymer volume fraction to under 10 Å at high volume fraction. At low polymer volume fraction and at high-q, the chains radius of gyration is given y R g = ξ = 113 Å and the end-to-end chain distance is R 1n = 6R g = 77 Å. 100 EO/D O, M w = 100,000 g/mole Correlation Length (Å) Y = M0*X M1 M M R o C EO Weight Fraction Figure : Variation of the correlation length with polymer volume fraction. 5. THE SINODAL TEMERATURE The correlation length ξ and the coefficient C increase with increasing temperature T due to increased composition fluctuations when approaching phase separation. The EO/dwater system is characterized y a lower critical solution temperature (LCST), i.e., it phase separates upon heating. The spinodal (phase separation) temperature T s is otained 3

4 when C diverges; it can e accurately estimated from the intercept of a C -1 vs T -1 plot of data taken at various temperatures. In this case of 1 % EO/d-water, one finds T s = 17 o C. 1.4 EO/D O, M w = 100,000 g/mole % 1/C T s = 17 o C Inverse Temperature (K -1 ) Figure 3: Variation of the high-q inverse intensity C -1 with inverse temperature T -1. The intercept represents the spinodal (phase separation) oundary line. 6. THE EXCLUDED VOLUME ARAMETER Our fitting results (high-q orod exponent) for the 1 % EO/d-water solution show that chains are mostly swollen at low temperatures (m = 1.85 which corresponds to an excluded volume parameter around ν = 1/m = 0.54) and change to theta conditions at high temperatures (m =.0 which corresponds to an excluded volume parameter around ν = 0.5) as the spinodal temperature is approached. 4

5 .05 EO/D O, M w = 100,000 g/mole orod Exponent m % Temperature ( o C) Figure 4: Variation of the orod exponent with temperature. olymer chains change from a swollen state to a theta condition as the spinodal temperature is approached. 7. BRANCH OF THE HASE DIAGRAM The spinodal temperature T s was otained from the various EO volume fraction samples that were measured. A ranch of the LCST phase diagram was otained. What is interesting is that the phase oundary line T s is estimated through extrapolation (i.e., efore reaching it). For some of our samples, T s happens to e aove the oiling temperature of water (and therefore unreachale except when measurements are made inside a pressure cell). The SANS technique is a good monitor of phase separation ecause it is sensitive to composition fluctuations which get enhanced close to phase oundary lines. 5

6 50 EO/D O, M w = 100,000 g/mole Spinodal Temperature ( o C) Data and Interpolation Two-phase region One-phase region EO Volume Fraction (%) Figure 5: Limited ranch of the phase diagram for the EO/d-water polymer solution system. hase separation is otained upon heating (LCST ehavior). 8. OLYMER SOLUTION THERMODYNAMICS olymer solutions can phase separate upon heating (LCST ehavior) or upon cooling (UCST ehavior). olymers that dissolve in organic solvents tend to e characterized y a UCST whereas water-solule polymers tend to follow LCST thermodynamics. The Flory-Huggins approach is a mean-field theoretical model that predicts phase separation ehavior. This model will e discussed later for polymer lends. 9. THE ZERO AVERAGE CONTRAST METHOD The zero average contrast method (also called high concentration method) uses variation of the fraction of deuterated polymer and deuterated solvent ut keeping the total polymer concentration (or volume fraction) constant to measure the single-chain form factor even at high concentrations ecause the interchain contriution cancels out. A series of EO/water solutions were prepared wherey the total polymer fraction was kept constant (volume fraction of 4 %) ut the relative amount of deo/heo was 6

7 varied. In order to isolate the single-chain contriution, we used mixtures of D O and H O solvent molecules that match the average polymer scattering-length density in each case. For such heo/deo/h O/D O mixtures, the scattering intensity is given y: dσ(q) = dω ΔB = Δρ ({ ΔB } { ΔB } ) n v (Q) + { ΔB } n v (Q) D H { } + Δρ { Δ B } Δρ Δρ D H = = D = Δρ D D + Δρ H H D S ( ρ ρ ) = D S v D vs = v H S ( ρ ρ ) H S H vs H. S T () Here, H and D are the scattering lengths for the heo and deo monomers, v H and v D are the corresponding specific volumes, and H and D are the corresponding polymer volume fractions (and similarly for the solvent scattering length density S /v S ). In order to arrive at this formula, it was assumed that the protonated and deuterated polymer degrees of polymerization and specific volumes are matched. The degree of polymerization used here, n, represents the value for the two mixed polymer species (n H = n D = n ). The total polymer volume fraction ( = H + D ) and polymer specific volume v (v = v H = v D ) have also een defined. The single-chain form factor S (Q) and the total-chain structure factor (including intra-chain and inter-chain contriutions) T (Q) have also een defined. The average contrast match condition zeroes the second term in the cross section equation leaving only the first term proportional to S (Q). This formula assumes that deuteration does not affect chain structure or interactions. This is oviously an assumption for our hydrogen-onded system. 7

8 4 % heo/d O 4 % heo/h O 50 % deo / 50 % heo average contrast match line 63.5 % D O / 66.5 % H O 17. % D O / 8.8 % H O 4 % deo/d O 4 % deo/h O Figure 6: The series of EO/water solutions for the average contrast match series. Mixtures of deuterated and non-deuterated polymers (deo and heo) and solvents (D O and H O) are used. Specific values for the defined parameters for our system are as follows. n heo = 73, n deo = 15, (3) heo = 4.14*10-13 cm, deo = 45.78*10-13 cm, H O = -1.67*10-13 cm, D O = 19.14*10-13 cm, v heo = v deo = cm 3 /mol, v H O = v D O = 18 cm 3 /mol. The four possile contrast factors corresponding to the 4 corners in the figure are as follows: V V V V deo deo heo heo heo heo deo deo N = 9.66*10 mol / cm HO 3 4 av V (4) HO V V DO DO HO HO N N av av = 5.50*10 =.38* mol / cm mol / cm DO 5 4 N av = 7.53*10 mol / cm V. DO 4 4 8

9 Here, we have multiplied y Avogadro s numer (N av = 6.0*10 3 molecules/mol) for convenience. The strongest neutron contrasts correspond to the two mixtures: deo/h O and heo/d O. Contrasts corresponding to the other two mixtures are much lower. The higher incoherent ackground is found in the samples with the most hydrogen (i.e., with non-deuterated solvent). Data from a specific 4 % EO/water mixture with 50 % deo/50 % heo and 63.5 % D O / 36.5 % H O are shown in a figure. This mixture is represented y a circle on the average contrast match line in the same figure. This sample is characterized y the singlechain scattering feature only. The low-q feature representing clustering has mostly disappeared. This method is useful for isolating single-chain properties in semi-dilute (and even concentrated) polymer solutions. 10 Scattering Intensity (cm -1 ) 1 SANS Data 4% EO/water, T=10 o C 50 % deo/50 % heo 63.5 % D O/36.5 % H O Scattering Variale Q (Å -1 ) Figure 7: SANS data from a EO/water sample on the average contrast match line. Mixtures of deo/heo and D O/H O are used to cancel out scattering from the clusters leaving scattering from single polymer chains only. M w = 100,000 g/mol for oth the deo (deuterated) and heo (non-deuterated) polymers. Nonlinear least-squares fit of this data to the Gaussian chain model with excluded volume (descried efore) gave a segment length of a EO = 6.7 Å and an excluded volume parameter of ν = Based on these numers, the radius of gyration can e estimated as 9

10 0.51 a EOn EO R g = = 137 Å. The EO chain degree of polymerization n EO = 00 (ν + 1)(ν + ) has een used. Fit to the simple empirical model form (descried efore) gave a correlation length ξ = 9.8 Å and a orod exponent m =.06. The radius of gyration can e estimated here also as R g = ξ = 131 Å. This shows acceptale agreement. REFERENCES. Flory, rinciples of olymer Chemistry, Cornell University ress, Ithaca, B. Hammouda, D. Ho, and S. Kline, SANS from oly(ethylene Oxide)/Water Systems, Macromolecules, 35, (00). B. Hammouda and D. Ho, Insight into Chain Dimensions in EO/Water Solutions, J. olym. Sci., olym. hys. Ed. 45, (007). QUESTIONS 1. What is the high-q expansion of the Deye function (form factor for Gaussian coil)?. What standard plot is used to otain the radius of gyration, the correlation length, the persistence length? 3. What is the meaning of the correlation length? 4. What does it mean to refer to the EO/water solution as characterized y an LCST phase diagram? 5. A orod exponent of 5/3 is an indication of what type of polymer chains? 6. The high-q SANS data is characteristic of what type of interactions in polymer solutions? ANSWERS 1. The high-q expansion of the Deye function is: Lim (Q ) = /(Q R ).. The Guinier plot is used to otain the radius of gyration, the Zimm plot is used to otain the correlation length, and the Kratky-orod plot is used to otain the persistence length. 3. The correlation length is the average distance etween entanglement points. 4. The EO/water solution is characterized y a Lower Critical Solution Temperature phase diagram means that phase separation occurs upon heating. 5. A orod exponent of 5/3 is an indication of fully swollen polymer chains. 6. The high-q SANS data is characteristic of solvent/polymer interactions (the so-called solvation shell) and therefore of the thermodynamics of mixing. g 10